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Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov Analysis Rush D. Robinett, III David G. Wilson Energy, Resources & Systems Analysis Center Sandia National Laboratories, P.O. Box 5800 Albuquerque, NM, 87185-1108 USA [email protected] [email protected] 47 th IEEE Conference on Decision and Control Fiesta Americana Grand Coral Beach Hotel Cancun, Mexico December 8, 2008 CDC ’08 Workshop Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
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Page 1: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Nonlinear Power Flow Control Design:Utilizing Exergy, Entropy, Static and Dynamic

Stability, and Lyapunov Analysis

Rush D. Robinett, III David G. WilsonEnergy, Resources & Systems Analysis Center

Sandia National Laboratories, P.O. Box 5800

Albuquerque, NM, 87185-1108 USA

[email protected] [email protected]

47th IEEE Conference on Decision and Control

Fiesta Americana Grand Coral Beach Hotel

Cancun, Mexico December 8, 2008

CDC ’08 Workshop

Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed

Martin Company, for the U.S. Department of Energy’s National Nuclear Security Administration under

contract DE-AC04-94AL85000.

Page 2: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Goal of Workshop

• Present innovative control system design process based on R&D at SNL in the area of renewable energy electric grid integration (future electric power grid control, wind turbine load alleviation control, and novel micro-grid control design).

• The main result of this research is a unique set of criteria to design controllers for nonlinear systems with respect to both performance and stability, instead of following the typical linear controller zero-sum design trade-off process between stability and performance.

• This control design process combines concepts from thermodynamic exergy and entropy; Hamiltonian systems; Lyapunov's direct method and Lyapunov optimal analysis; static and dynamic stability analysis, electric AC power concepts including limit cycles; and power flow analysis. The thermodynamic concepts combined with Hamiltonian systems provide the theoretical foundations necessary to view control design as power flow control problem of balancing the power flowing into the system versus the power being dissipated within the system subject to power being stored in the system.

• Emphasis is placed on necessary design steps for which the concepts are introduced and explained with examples and case studies.

Page 3: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Agenda

8:30 - 9:15 Welcome and Introduction Charlie Hanley

9:15 - 10:15 Theory Rush Robinett III

10:15 - 10:30 BREAK!

10:30 - 12:00 Continue Theory/Discussions Rush Robinett III

David Wilson

12:00 - 1:00 Lunch!

1:00 - 2:30 Case Studies 1 & 2 Rush Robinett III

David Wilson

2:30 - 2:45 BREAK!

2:45 - 4:15 Case Studies 3 & 4 Rush Robinett III

David Wilson

4:15 - 4:30 Open Discussions All

4:30 Adjourn!

Page 4: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Speakers

Energy Infrastructure Futures - Sandia National Labs

• Charles J. Hanley (Keynote Speaker) - Mr. Hanley is manager of Sandia’s Photovoltaic Systems Evaluation Laboratory and the Distributed Energy Technologies Laboratory. He manages research and development focused on new photovoltaic components and systems with improved performance, higher reliability, and lower overall cost, in support of the US Department of Energy’s efforts to make these technologies cost effective. This research includes; modernization of the electric grid through controls and communications, to support the eventual high penetration of renewable energy technologies into our infrastructure. Until 2002, Charlie managed Sandia’s international renewable energy programs, through which he oversaw the implementation of more than 400 photovoltaic, wind, and passive solar energy systems in Latin America. He received his B.S. degree in Engineering Science from Trinity University in San Antonio, TX, and his M.S. degree in Electrical Engineering from Rensselaer Polytechnic Institute, in Troy, N.Y.

• Rush D. Robinett III - has three degrees in Aerospace Engineering from Texas A&M University (B.S. - 1982, Ph.D.- 1987) and The University of Texas at Austin (M.S. - 1984). He has authored over 100 technical articles including two books and holds 7 patents. He began working on ballistic missile defense at Sandia National Laboratories in 1988. In 1995, he was promoted to Distinguished Member of Technical Staff. In 1996, he was promoted to technical manager of the Intelligent Systems Sensors and Controls Department within the Robotics Center. In 2002, he was promoted to Deputy Director of the Energy and Transportation Security Center where he is developing distributed power and transportation infrastructures focusing on exergy, entropy and information metrics.

• David G. Wilson - has three degrees in Mechanical Engineering from Washington State University (B.S. - 1982, M.S. 1984) and The University of New Mexico (Ph.D. - 2000). He has authored over 50 technical articles including two books. He is currently a Principal Member of Technical Staff at Sandia National Laboratories, Energy Systems Analysis Department. He has over 20 years of research and development engineering experience in energy systems, robotics, automation, and space and defense projects. His current areas of research include: design, analysis, and implementation of nonlinear/adaptive control; distributed decentralized control architectures; exergy/entropy and collective control for dynamical systems.

Page 5: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Workshop Notes and References

Viewgraph Handouts (booklet)

Several conference and journal papers included:1) R.D. Robinett III and D.G. Wilson, What is a Limit cycle?, International Journal of Control, Vol.

81, No. 12, Dec. 2008, pp. 1886-1900.

2) R.D. Robinett III and D.G. Wilson, Exergy and Irreversible Entropy Production Thermodynamic Concepts for Nonlinear Control Design, accepted for publication, International Journal of Exergy, February 2008.

3) R.D. Robinett III and D.G. Wilson, Collective Plume Tracing: A Minimal Information Approach to Collective Control, accepted for publication, International Journal of Robust and Nonlinear Control, Nov. 2008.

4) R.D. Robinett III and D.G. Wilson, Nonlinear Power Flow Control Applied to Power Engineering, SPEEDAM 2008, International Symposium on Power Electronics, Electrical Drives, Automation and Motion, Ischia, Italy, June 2008.

5) R.D. Robinett III and D.G. Wilson, Collective Systems: Physical and Information Exergies, SNL, SAND2007-2327 Report, March 2007.

6) R.D. Robinett III and D.G. Wilson, Exergy and Entropy Thermodynamic Concepts for Control System Design: Slewing Single Axis, AIAA Guidance, Navigation, and Control Conference and Exhibit, Keystone, Co., August 2006.

7) R.D. Robinett III and D.G. Wilson, Exergy and Entropy Thermodynamic Concepts for Nonlinear Control Design, Proceedings of IMECE2006-15205, 2006 ASME International Mechanical Engineering Congress and Exposition, Nov. 2006, Chicago, Illinois.

8) R.D. Robinett III and D.G. Wilson, Exergy and Irreversible Entropy Production Thermodynamic Concepts for Control System Design: Robotic Servo Applications, Proceedings of the 2006 IEEE International Conference on Robotics and Automation, Orlando, Florida, May 2006.

9) Robinett, III, R.D., Wilson, D.G., and Reed, A.W., Exergy Sustainability for Complex Systems, No. 1616, InterJournal Complex Systems, New England Complex Systems Institute, 2006.

Page 6: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Acknowledgments

• Ben Schenkman, Dept. 6336 – Power Engineering Modeling

•Val Weekly, Dept. 6330 – Workshop Coordinator

• Innovative Control of a Flexible, Adaptive Energy Grid, Sandia

National Laboratories, LDRD Project, No. 09-1125, Oct. 2006 –

Oct. 2009.

Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed

Martin Company, for the U.S. Department of Energy’s National Nuclear Security Administration under

contract DE-AC04-94AL85000.

Page 7: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Outline

• Theory (Morning)

– Thermodynamics

– Mechanics

– Stability

– Advanced Control Design

• Case Studies (Afternoon)

– Control Design Issues

– Collective Plume Tracing

– Nonlinear Aeroelasticity

– Power Engineering

Page 8: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

THERMODYNAMICS

• First Law - Energy

• Second Law – Stability/Entropy

• Equilibrium Thermodynamics - Reversible/Irreversible Paths

• Local Equilibrium – Non Equilibrium Thermodynamics

• Rate Equations – Energy, Entropy, and Exergy

Page 9: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Thermodynamics (cont.)

• First Law [Ref. Gyftopoulos] - A statement of the existence of a

property called energy

– Energy is a state function: path independent; E

– Adiabatic work is path independent

Energy at state i of system

Work done by system between

states 1 and 2

−iE

−12W

1212 WEE −=−

[Ref. Gyftopoulos]: E.P. Gyftopoulos and G.P. Beretta, Thermodynamics,

Foundations and Applications, MPC, NY, 1991.

Page 10: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Thermodynamics (cont.)

- Corollary of First Law

change of state; path independent

flow of heat; path dependent

work done by the system; path dependent

(conservation of energy)

WQdE δδ −=

−dE

−Qδ

−Wδ

Page 11: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Thermodynamics (Cont.)

• Second Law – [Ref. Gyftopoulos]: A statement of the existence of stable equilibrium states and of special processes that connect these states together

– Equilibrium State: A state that does not change with time while the system is isolated from all other systems in the environment

– Stable Equilibrium State: A finite change of state cannot occur,regardless of interactions that leave no net effects in the environment

– Restate Second law: Among all the allowed states of a system with given values of energy, number of particles, and constraints, one and only one is a stable equilibrium state

[Ref. Gyftopoulos]: E.P. Gyftopoulos and G.P. Beretta, Thermodynamics,

Foundations and Applications, MPC, NY, 1991.

Page 12: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

– Reversible and Irreversible Processes: A process is reversible if the system

and its environment can be restored to their initial states, except for

differences of small order of magnitude than the maximum changes that occur

during the process

Example 1: Expansion of Gas

Thermodynamics (cont.)

Reservoir, Reservoir,oT

FF

oT

Reversible Expansion

Adiabatic Expansion – Irreversible

Page 13: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Thermodynamics (cont.)

– Available work:

– Entropy:

– Adiabatic processes: Reversible:

Irreversible:

Ω

S

12121212 )(;)( WWWrevrev ≥−=Ω−Ω

)( Ω−Ε≡ ddcdS R

0)( =REVdS

0)( ≥irrevdS

Page 14: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Thermodynamics (cont.)

– Corollary: At a stable equilibrium state, the entropy will be at its maximum value for fixed values of energy, number of particles, and constraints

– Reversible process: Interacting systems quasi-statically pass only through stable equilibrium states

;)( TdSQdQ REV == δ

;T

dQdS =

etemperaturT −

∫==T

dQS 0

Page 15: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Thermodynamics (cont.)

– Irreversible process [Ref. Prigogine]:

- Entropy change due to exchange of energy and matter

(entropy flux)

- Entropy change due to irreversible processes

T

QdS

δ≥

SdSddS ie +=

Sde

Sd i

[Ref. Prigogine] D. Kondepudi and I. Prigogine, Modern Thermodynamics: From

Heat Engines to Dissipative Structures, John Wiley & Sons, New York, N.Y., 1999.

Page 16: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Thermodynamics (cont.)

Sdi

Sde

Figure 1- Entropy changes

0≥=∑ kk

k

i dXFSd

th

k kF − thermodynamics force

thermodynamics flowth

k kX −

Page 17: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Thermodynamics (cont.)

• Equilibrium Thermodynamics [Ref. Prigogine] – In classical

thermodynamics it is assumed that every irreversible transformation that

occurs in nature can also be achieved through a reversible process where

∫+=2

112

T

dQSS

irrev

1

2

rev

E

S

Reservoir

T=Constant

[Ref. Prigogine] D. Kondepudi and I. Prigogine, Modern Thermodynamics: From

Heat Engines to Dissipative Structures, John Wiley & Sons, New York, N.Y., 1999.

Page 18: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Thermodynamics (cont.)

VpdTdS

VpdTdSdU

dWVpd

dQTdS

dEdU

VpdTdSdU

∫∫∫∫∫

=⇒

−==

=

=

=

−=

0

4

3 2

1Work Done (area)

T

1

23

4

S

p

- A uniform temperature exists throughout the system.

= internal energy

volumeVpressurep −− ,;

- Reversible Cycle Processes:

V

Page 19: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Thermodynamics (cont.)

• Local Equilibrium – Thermodynamic quantities are well-defined concepts locally (i.e., within each elemental volume)

– Temperature is not uniform, but is well defined locally

– Non-equilibrium systems: define thermodynamic quantities in terms ofdensities

– Thermodynamic variables become functions of position and time

• Rate Equations

– Energy:

– Entropy:

– Exergy (Available work)

...)( +++++= ∑∑∑ llll

l

k

k

j

j

pekehmWQE &&&&

0≥=

++=+=

∑∑

ll

l

i

ikk

kj

j

j

ie

XFS

SsmT

QSSS

&&

&&&

&&&

io

l

Flow

llok

k

j

j

o

j

o STmVpWQT

TSTE &&&&&&&& −+−+−=−=Ξ ∑∑∑ ζ)()1(

Page 20: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

MECHANICS

• Work, Energy, and Power

• Energy Diagrams and Phase Planes

• Hamiltonian Mechanics

• Connections between Thermodynamics and Hamiltonian Mechanics

• Line Integrals and Limit Cycles

Page 21: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

• Work, Energy, and Power

– Work:

Work

Force Vector

Position Vector

– Power:

Power

Velocity Vector

– Kinetic Energy: (Newton’s 2nd law)

rdFdW ⋅=−W

−F

−r

rFrdt

dFW

dt

dP &⋅=⋅==

−r&−P

rmF &&=

−⋅=

=

⋅=⋅=⋅=⋅=

rrmT

dTrrmdrdrmrdrmrdFdW

&&

&&&&&&

2

1

2

1

Kinetic Energy

Page 22: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

– Potential Energy:

Conservative and path independent force field

⋅=−

−=⋅

∫V

rdrFrVrV

rdrF

r

r

2

1

)()()(

)(

21

Potential Energy

==+⇒

−=⋅=

=⋅∫

E

EVT

rFdt

dT

rdF 0

Constant

)(rdV

⇒dt

dV0)( =+VT

dt

d

Total Energy (stored energy)

Page 23: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

– Time Dependent Potential Field: −=⋅ rdtrF ),(

t

VVT

dt

d

∂=+ )(

),( trdV

Page 24: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

-Non-conservative Forces:

Conservative Force (Potential Field)

Non-Conservative Force

NCc FFF +=

−cF

−NCF

rFt

VVT

dt

d

dt

dENC

&⋅+∂

∂=+= )(

Page 25: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

• Energy Diagrams and Phase Planes

– Mass, Spring, Damper System

2

2

1xmT &=

2

2

1kxV =

xcuF &−=

−u Control

Force

−xc& Damping

Force

k

c

xcukxxm &&& −=+

Page 26: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

a) Set conservative system, :oxcu == & =+= VTE Constant

2

2

1kxV =

x

2

2

1xmT &=

x&

xx

E E

E

Page 27: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

The system path is constrained to the energy storage surface along

a constant energy orbit

22

2

1

2

1oo kxxmE += &

E= constant

x&

x

Page 28: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

b) Set and

The system path is constrained to the energy storage surface as it

spirals down into the minimum energy state

0=u 0>c

Page 29: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

• Hamiltonian Mechanics

– Lagrangian:

Time Explicitly

N Dimensional Generalized Coordinate Vector

N Dimensional Generalized Velocity Vector

– Equations of Motion:

Generalized Force Vector

– Hamiltonian:

),(),,( tqVtqqTL −= &

−t−q

−q&

Qq

L

q

L

dt

d=

∂−

∂)(

&

−Q

),,(),,(1

tqqHtqqLqq

LH i

i

N

j

&&&&

=−∂

∂≡ ∑

=

Page 30: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

– Canonical Momentum:

– Hamilton’s Canonical Equations of Motion:

– Time Derivative of the Hamiltonian:

),,(),,(1

tqqLqptpqH ii

N

i

&& −= ∑=

j

j

j

j

j

Qq

Hp

p

Hq

+∂

∂−=

∂=

&

&

t

LqQ

qq

Lq

q

L

t

LqpqpH

jj

N

j

j

j

j

j

jjjj

N

j

∂−=

∂−

∂−

∂−+=

=

=

&

&&&

&&&&&&

1

1

)(

i

iq

Lp

&∂

∂=

Page 31: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

– For most natural systems:

Power (work/energy) flow equation

jj

N

j

qQqqH

t

L

&&& ∑=

=

=∂

1

),(

0

Page 32: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

• Connections between Thermodynamics and Hamiltonian Mechanics

– Conservative Mechanical Systems:

a) and Constant

b) Conservative Force (storage device)

For any closed path

0=H& =H

0=⋅=⋅=⋅ ∫∫∫ dtqQdtrFrdF &&

Page 33: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

– Reversible Thermodynamic Systems:

– Irreversible Thermodynamic Systems:

for

then and

T

QS

dtSSSdSddS

T

dQdS

T

dQdS

e

ieie

&&

&&

=⇒

=+=+=

==

=

∫∫∫

∫∫0][][

0

(Second Law)and 0=iS&

0][ =+= ∫∫ dtSSdS ie&&

0≤eS& 0≥iS

&

Page 34: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

– Hamiltonian is stored exergy since potential and kinetic energies are available work; Exergy rate relationship:

a) Power flowing in (N generators)

b) - Power Dissipation (M Dissipators)

c) (Assuming Local Equilibrium)

qFqQqQSTW

qFqQH

NCll

NM

Nl

jj

N

j

io

NCkk

k

&&&&&

&&&

⋅=−=−=Ξ

⋅==

∑∑

∑+

+== 11

−W&

ioST &

01

≥== ∑∑ kk

ko

kk

k

i qQT

XFS &&&

Page 35: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

d) Assumptions applied to exergy rate equation:

e) A conservative system is equivalent to a reversible system when

and

then and

0

0

01

0

=

=

≅−

∑ FLOW

kk

k

o

j

o

i

m

Vp

T

T

Q

ζ&

&

&

0=H& 0=eS&

0=iS& 0=W&

Page 36: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

• Line Integrals and Limit Cycles

– Cyclic Equilibrium Thermodynamics:

– Cyclic Non-Equilibrium Thermodynamics with Local Equilibrium

TdSdW

TdSVpd

dWTdSdE

VpdTdSdU

∫∫∫∫∫∫∫∫∫∫

=⇒

=⇒

=−=

=−=

0

0

dtqQdtqQ

dtSTdtW

dtSTWdtH

kk

NM

Nk

jj

N

j

i

io

][][

0][

11

0

&&

&&

&&&

∑∫∑∫

∫∫∫∫

+

+==

=⇒

=⇒

=−=

Page 37: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

– Sorting Power Terms:

a) Conservative Terms

Potential functions

b) Generator Terms

c) Dissipator Terms

⇒=⋅∫ 0dtqF c&

0][01

>∫⇒>⋅∫ ∑=

dtqQdtqF jj

N

j

cNCc&&

0][01

<∫⇒<⋅∫ ∑+

+=

dtqQdtqF kk

MN

Nk

cNCc&&

Page 38: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

– Limit Cycles:

dtqQqQ

dtSTdtW

kk

MN

Nk

jj

N

j

io

=

=

∑∫∑∫

∫∫+

+==

&&

&&

11

Page 39: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

van der Pol Oscillator: Limit Cycle Identification

-Dissipative

-Neutral

-Generative

-Hamiltonian

surface

Page 40: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Mechanics (cont.)

Linear Limit Cycles: Power Grid Applications

• Goal Power Engineering maximize real power flow: match frequency and phase of applied voltage and resulting current

• Exergy/Entropy Control Design identifies stability boundaries and improves system performance with nonlinear feedback by providing robustness to disturbances and adaptation to parameter changes

• Current Power Engineering solutions “open loop” –subject to changing load variations and delays due to manual operator set point updates and approvals

• Simple RLC circuit – Driven with 60 Hz sinusoidal input demonstrates variations and changes in load and frequency

Exergy/Entropy Distributed Control: Predicts

Performance and Stability for Power System

Page 41: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

STABILITY

• Static and Dynamic Stability

• Eigenanalysis

• Lyapunov Analysis

• Energy Storage Surface and Power Flow: Necessary and Sufficient

Conditions for Stability

Page 42: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

• Static and Dynamic Stability

– Static Stability: If the forces and moments on a body caused by a disturbance tend initially to return (move) the body towards (away from) its equilibrium state, the body is statically stable (unstable) [Refs. Anderson, Robinett]

a) Equilibrium State- An unaccelerated motion wherein the sums of the forces and moments on the body are zero

b) Neutral Stability - Occurs when the body is disturbed and the sums of the forces and moments on the body remain zero

[Ref. Anderson] J.D. Anderson, Jr., Introduction to Flight, McGraw-Hill, 1978.

[Ref. Robinett] R.D. Robinett III, A Unified Approach to Vehicle Design, Control, and

Flight Path Optimization, PhD Dissertation, Texas A&M University, 1987.

),( ee xx&

Page 43: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

Statically

Page 44: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

c) The Energy Storage Surface is

d) A system is statically stable if

for

Statically unstable if

And Neutrally stable if

0)( >xV

0),(

0

=

>

ee xxH

H

&

0)( <xV

0)( =xV

EVTH =+=

ee xxxx && ≠≠∀ ,

Page 45: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

• Dynamic Stability: Defined in terms of the time history of the motion

of a body after encountering a disturbance

– A body is dynamically stable (unstable) if, out of its own accord, it

eventually returns to (deviates from) and remains at (away from) its

equilibrium state over a period of time [Refs. Anderson, Robinett1]

a) A dynamically neutral stable body occurs when a limit cycle

exists [Refs. Robinett2, Robinett3]

b) A dynamically stable body must always be statically stable, but

static stability is not sufficient to ensure dynamic stability [Refs.

Abramson, Anderson, Robinett1] Therefore, static stability is a

necessary condition for stability

[Ref. Anderson] J.D. Anderson, Jr., Introduction to Flight, McGraw-Hill, 1978.

[Ref. Robinett1] R.D. Robinett III, A Unified Approach to Vehicle Design, Control, and Flight Path Optimization, PhD

Dissertation, Texas A&M University, 1987.

[Ref. Robinett2] R.D. Robinett III, What is a Limit Cycle?, Int’l Journal of Control, Vol. 81, No. 12, Dec. 2008, pp.

1886-1900.

[Ref. Robinett3] R.D. Robinett III and D.G. Wilson, Collective Systems: Physical and Information Exergies, Sandia

National Laboratories, SAND2007-2327 Report, March 2007

[Ref. Abramson] H.N. Abramson, Introduction to the Dynamics of Airplanes, Ronald Press, 1958.

Page 46: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

c) The time derivative of the energy storage surface defines the power

flow into, dissipated within, and stored in the system

d) A system is dynamically stable if

dynamically unstable if

and dynamically neutral stable if (Limit Cycle)

01

<= ∫c

oc

AVE dtHHτ

τ&&

01

>= ∫ dtHHc

oc

AVE&&

τ

τ

01

== ∫ dtHHc

c

AVE&&

ττ

Page 47: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

• Eigenanalysis: The goal is to relate/ extend eigenanalysis of linear

systems to nonlinear systems via the energy storage surface, power flow,

and limit cycles

– Mass, spring, damper system

);(2

1

)()(

2 xVxmEH

uxfxgxm

+==

+−=+

&

&&&

x

xVxg

∂=

)()(

Page 48: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

a) Linearized system

b) Assume , Eigenvalue Problem

Undamped natural frequency (eigenvalue) of a statically stable and

dynamically neutral stable system

xxcuxkxxmH

uxckxxm

kxxmH

kcmxcxfkxxV

&&&&&&

&&&

&

&&

][][

02

1

2

1

0,,;)(;02

1)(

22

2

−=+=

+−=+

>+=

>=>=

m

k

kxxm

=

=+

2

0

ω

&&

0,0 === Hxcu &&

Page 49: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

c) Assume and

This system is statically stable and dynamically stable if

Dynamically unstable

Dynamically neutral stable if

d) Extend results to nonlinear systems:

2

22

][][

02

1

2

1

xKcxkxxmH

kxxmH

D&&&&&

&

+−=+=

>+=

xKu D&−= 0>c

0>+ DKc0<+ DKc

0=+ DKc

04

1

2

1

2

1)(

2

1 4222 >++=+= xkkxxmxVxmH NL&&

Page 50: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

This system is statically stable and dynamically neutral stable (limit cycle) if

with nonlinear frequency content of

and dynamically stable if

and dynamically unstable if

0)]([1

)]([1

=−=+= ∫∫ dtxxfudtxxgxmHcc

cc

AVE&&&&&&

ττ ττ

0)( =+ xgxm &&

0)]([1

<−= ∫ dtxxfuHc

oc

AVE&&&

τ

τ

0)]([1

>−= ∫ dtxxfuHc

oc

AVE&&&

τ

τ

Page 51: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

• Lyapunov Analysis: The goal is to relate static and dynamic stability, energy storage surface, and power flow to Lyapunov Analysis

– The definition of static stability is equivalent to the requirement placed on a Lyapunov function: must be positive definite

– A statically unstable system is unstable in the sense of Lyapunov.

Assume and for pick Lagrangian

for which is unstable

0),( >= Hxx &VVVV

0== xcu & 2

2

1)( kxxV −= ;0>k

02][

02

1

2

1 22

>=+=

>+=−==

xkxxkxxm

kxxmVTL

&&&&&

&

VVVV

VVVV

0=−kxxm&&

Page 52: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

– The definition of dynamic stability is equivalent to the requirements

placed on a Lyapunov function. A system is asymptotically stable if

[Ref. Robinett1]

– And unstable if

The limit cycle defines the stability boundary between asymptotically

stable and unstable [Ref. Robinett2]

( ) 0)]([1

0

<−=

>=

∫ dtxxfu

H

c

oc

ave&&&

τ

τVVVV

VVVV

( ) 0)]([1

0

>−=

>=

∫ dtxxfu

H

c

oc

ave&&&

τ

τVVVV

VVVV

[Ref. Robinett2] R.D. Robinett III and D.G. Wilson, What is a Limit Cycle?,

International Journal of Control, Vol. 81, No. 12, Dec. 2008, pp. 1886-1900.

[Ref. Robinett1] R.D. Robinett III and D.G. Wilson, Exergy and Irreversible Entropy

Production Thermodynamic Concepts for Nonlinear Control Design, accepted for

publication, International Journal of Exergy, Feb. 2008.

Page 53: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

• Energy Storage Surface and Power Flow: Necessary and Sufficient

Conditions for Stability

– Energy Storage Surface: Total energy; stored exergy;

paths/trajectories are constrained to this surface; infrastructure

a) Determines static stability which is a necessary condition for

stability

b) Proportional feedback affects static stability; assume

for and 42

2

1

2

1)( xkkxxV NC+−= 0, >NLkk =u

422

3

3

4

1][

2

1

2

1

0][

xkxkKxmH

xkxkKxm

xKuxkkxxm

NCp

NLp

pNL

+−+=

=+−+

−==+−

&

&&

&&

,xK p− 0=c

Page 54: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

pKk >

Page 55: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

• Power Flow: Exergy Flow; 3 types– into, dissipated within, and stored in

the system; determines the path/trajectory across the energy storage

surface

a) Determines the dynamic stability which is a sufficient condition

for stability

b) Sort Power Terms:

c) Derivative Feedback:

d) Integral Feedback:

iSTWHo

&&& −=

2xKST Dio&& =⇒

xKu D&−=

xdtKu I ∫−=

xxdtKW I&& ∫−=⇒

Page 56: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

,0>c dtSTdtW iooocc && ττ ∫<∫

xckxxm &&& −=+

Dynamically Stable

Page 57: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

,0>c dtSTdtW iooocc && ττ ∫>∫

Dynamically Unstable

xckxxm &&& =+

Page 58: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Stability (cont.)

,0=c dtSTdtW iooocc && ττ ∫=∫

Dynamically Neutral Stable

0=+ kxxm &&

Page 59: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

ADVANCED CONTROL DESIGN

•Distributed Parameter/PDE’s

•Fractional Calculus

•Optimal

•Robust/ Tracking

•Adaptive/Tracking

Page 60: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

• Distributed Parameter/PDE’s

– Vibrating String

x)( 1txw

0 L

Tension in String

Mass/unit length

=τ=)(xσ

t

ww

∂=&

x

wwx

∂=

0),(),0( == tLwtw

Page 61: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

a)

b) Non-conservative Distributed Force

dxwV

dxwxT

x

L

L

2

0

2

0

2

1

)(2

1

=

=

τ

σ &

workwdxtxFWL

==⇒ ∫ ),(0

Page 62: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

c) Extended Hamilton’s Principle

0

0

)(2

1

][

0 0

22

=−+

=

∂−

∂−

=

+−=

+−Τ=Ι

∫ ∫

∫∫

wwF

w

f

tw

f

xw

f

fdxdt

dxdtFwww

dtWV

xx

tx

t L

x

L

o

t

o

t

o

&&

&

στ

τσ

;Fww xx =−⇒ τσ && 0),(),0( == tLwtw

Page 63: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

d) Static Stability

for

e) Dynamic Stability

0)(2

1 22 >+=+= ∫ dxwwVTH x

L

oτσ &

0, >τσ

dxwF

dxwww

dxwwwwH

L

o

xx

L

o

xx

L

o

&

&&&

&&&&&

=

−=

+=

][

][

τσ

τσ

;01

=

=⇒ ∫∫ dtdxwFH

L

oc

AVEc

&&ττ

Limit Cycle

Page 64: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

• Controller: wKwdtKwKF DIxxp&−−= ∫

dxwwKwwH

wKwdtKwKw

dxwKwVTH

xxp

L

c

DIxxp

xp

L

c

])([

)(

0])([2

1

0

22

0

&&&&&

&&&

&

++=

−−=+−

>++=+=

τσ

τσ

τσ

dxwwKwdtK

dxwwKw

DI

L

xxp

L

&&

&&&

][

])([

0

0

−−=

+−=

∫∫

∫ τσ

Page 65: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

• Fractional Calculus

– Differintegral: Derivative and Integral to Arbitrary order

a) Oldham and Spanier [Ref. Oldham]:

q= Arbitrary real number

Gamma Function

b) Podlubny [Ref. Podlubny]:

−+Γ

−Γ

−Γ

∞→=

−∑

=

])[()1(

)(

)(

][lim

)]([

1

0 N

axjxf

j

qj

q

N

ax

Naxd

fd N

j

q

q

q

=+Γ )1( j

!

)1()2)(1()(

)()()1(lim)(

r

rqqqqq

r

rhtfhtfD q

r

rn

or

q

oh

q

ta

atnh

+−⋅⋅⋅−−=

−−= ∑=

→−= h t== small change in

[Ref. Podlubny] I. Podlubny, Fractional Differential Equations, Academic Press, N.Y., 1999.

[Ref. Oldham] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, N.Y., 1974.

Page 66: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

c) First Order Fractional Difference approximation

for

and for

d) Inhomogeneous Bagley-Torvik Equation

)()(~ )(

]/[

khtfwhtfD q

k

ht

ok

qq

to −= ∑=

−=

k

qw q

k )1()(

,...2,1,0=k

1)( =q

ow)(

1

)( )1

1( q

k

q

k wk

qw −

+−= ,...3,2,1=k

0),()()()( 2/3 >∀=++ ttutCytyDBtyA to&&

0)0()0( == yy &

Page 67: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

e) Generalized to

f) First-Order Approximation/numerical algorithm

For

0),()()()( >∀=++ ttutKxtxDKtxM q

toq&&

0)0()0( == xx &

01 == xxo

)2(

)(

1

)2(

)2(

2112 )2()(q

q

jm

q

j

m

j

q

q

q

q

mmmmm

hKM

xwhK

hKM

xxMKxuhx

−=

−−−−

+

++

−+−=

,...3,2=m

Page 68: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

g) Sort power terms:

xxKxdtK

xxKKxMH

xKKxMH

xKxdtKxKu

uKxxCxM

DI

p

p

DIp

&&

&&&&

&

&

&&&

][

])([2

1

0)(2

1 22

−∫−=

++=

>++=

−∫−−=

=++

Page 69: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

Storage: and

Dissipator:

Generator:

2

2

dt

xdo

o

dt

xd

1

1

dt

xd

1

1

dt

xd

:q

Storage Dissipator Storage Generator

2 1 0 -1

Type:

Coupled

)21(

Page 70: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

h) Lag – Stabilized Controller [Ref. Robinett]

For

Equivalent to positive proportional and negative derivative

feedback

Shifted Sinusoids

)(txDKu q

toqq −=

:2,0=q

:02 >> q

:10 −≥> q

Storage

Dissipator

Generator

)()( dp txKtuKxxM τ−==+&&

0, >dpK τ

⇒xKxKu Dp&−≅

[Ref. Robinett] R.D. Robinett III, B.J. Peterson, J.C. Fahrenholtz, Lag-Stabilized Force Feedback

Damping, Journal of Intelligent and Robotic Systems, Vol. 21, Issue 3, March 1998, PP. 277-285.

Page 71: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

The Fractional Calculus Approach

• Fractional calculus compared with PID type controllers

q=α

0 5 10 15 20-4

-2

0

2

4

6

8

Time (sec)

Position (m)

yPD Control

yFC with α=1.0

0 5 10 15 20-8

-6

-4

-2

0

2

4

6

8

Time (sec)

Position (m)

yP Control

yFC with α=0.0

0 5 10 15 20-25

-20

-15

-10

-5

0

5

10

15

20

Time (sec)

Position (m)

yI Control

yFC with α=-1.0

0 5 10 15 20-4

-2

0

2

4

6

8

Time (sec)

Position (m)

yPD Control

yFC with α=0.5

0 5 10 15 20-20

-15

-10

-5

0

5

10

15

Time (sec)

Position (m)

yPI Control

yFC with α=-0.5

Page 72: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

• Optimal [Ref. Ogata] -

Asymptotically Stable to x=0

);,( uxfx =&

→= ))(,( xgxfx&

)],([min),(~

min

0),(),(~

),(

uxLuxH

uxLuxH

dtuxLJ

uu

o

+=

≥+=

= ∫∞

VVVV

VVVV

&

&

VVVV &= 0),(1

=+= uxLuu

1uu=⇒VVVV & ),( 1uxL−=

dttutxLxxo

))(),(())0((0))(( 1∫∞

−==∞ VVVVVVVV

dttutxLxo

))()),(())0(( 1∫∞

=VVVV

)(xgu =

Note:

[Ref. Ogata] K. Ogata, Modern Control Engineering, Prentice-Hall, Inc., N.J., 1970.

Page 73: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

a) Example #1: ukxxcxm =++ &&&

][2

1][

2

1 2222

oo kxxmkxxmHJ o +=+−=−= ∞ &&

dtxxcudtxkxxmoo

&&&&& ][][ −−=+−= ∫∫∞∞

22

2

1

2

1

0][

kxxm

xcuxxcu

+=

<⇒<−=

&

&&&&

VVVV

VVVV

Page 74: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

b) Example #2: ukxxm =+&&

])4([2

1

:0*

][2

1

2

1

),(

][

2/12

2

2

13

2

3

2,1

21

2

3

2

2

2

1

22

2

3

2

2

2

1

2

3

2

2

2

1

⟩+⟨−±−=∗

==

++=−

=+=⇒+=

−−−=−=

++= ∫∞

xkxkkxxk

u

kk

ukxkxkxu

xuxkxxmkxxm

ukxkxkuxL

dtukxkxkJo

&&&

&&

&&&&&&

&&

&

VVVVVVVV

VVVV

xk

uxuku &&3

3

1,0)( −==+

Page 75: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

– Fixed Final Time: [ ] dtuuxJft

o

22

2 )(2

1+= ∫

TTxfLHuxfxuI ),(;

~);,(; θθλτθ &&&& =+====

BuxAuIx

x

Iu

x

x

xx +=

+

=

=

=

=

/1

0

00

10

/ 2

12

2

1

θ

θ&&

&

&

&&

)1(

1

.0

/~/

~

)1(/1

~

0

2

1)(

2

1~

2

2

22

2

12

2

2

11

12

2

2

1

2

1

2

222

2

2

221

22

2

0

+−

==

−−=

==⇒

−−=

∂∂−

∂∂−=

+−=⇒++=∂

∂=

+++=

xIu

Ix

xu

const

xuxH

xH

xIuI

uuxu

H

I

uxuuxH

o

λ

λλ

λλ

λλ

λ

λλ

λλ

&

&&

&

τ

θ

I, inertia

Horizontal Link

torque,

angle,

Page 76: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

• Homotopy from i) min. energy to ii) hybrid min. energy plus min.

power, to iii) min. power cost functions

[ ]

BuxAx

dtuWxuWJft

o

+=

−+== ∫&

)()1())((2

1)( 2

2

2

21 ξαξαξφ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-3

-2

-1

0

1

2

3

4

Time (sec)

Min

imu

m E

ffo

rt T

raje

cto

rie

s (

θ,

θd

ot,

τ )

Horizontal Link Slew POWER opt

Angle

Angle Rate

Torque

0 0.2 0.4 0.6 0.8 1-4

-3

-2

-1

0

1

2

3

4

Time (sec)

Po

we

r (W

) T

orq

ue

(N

-m)

Horizontal Link Slew POWER opt

Power

Torque

0)()(

0)()(

0)()(

00

2

1

==

==Ψ

=−=Ψ

tt

t

t

f

desiredf

θθ

θξ

θθξ

&

&

Page 77: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

• Robust / Tracking

– Nonlinear Model:

a) Hamiltonian:

;)()( uxfxgxm +−=+ &&&

)()(2

1 2

RR xxVxxmH −+−= &&

[ ] )()()( RRR xxxxgxxmH &&&&&&& −−+−=

[ ]uuu

xxuxfxgxxgxm

REF

RRR

∆+=

−+−−−+−= )()()()( &&&&

);(ˆ)(ˆ)(ˆˆ xfxgxxgxmu RRREF&&& ++−−=

)()()(ˆ RDRIR xxKdtxxKxxgu && −−−∫−−−=∆

x

xVxg

∂=

)()(

x

xVxg

∂=

)(ˆ)(ˆ

Robustness/Tracking addressed in the RLC example

Page 78: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

– Static Stability:

– Dynamic Stability:

– Robustness:

0)(ˆ)()(2

1ˆ 2 >−+−+−= RRR xxVxxVxxmH &&

[ ] )()(ˆ)()(ˆRRRR xxxxgxxgxxmH &&&&&&

&−−+−+−=

[ ] )()()()(ˆ)( RRRR xxuxfxgxxgxxgxm &&&&& −+−−−+−+−=

[ ]⟩−⟨+⟩−⟨+⟩−−−⟨+⟩−⟨= )()(ˆ)()(ˆ)(ˆ)(ˆ xfxfxgxgxxgxxgxmm RRR&&&&

] )()()( RRDRI xxxxKdtxxK &&&& −−−−∫−

[ ] []

AVER

RRRRIAVERD

xxxfxf

xgxgxxgxxgxmmdtxxKxxK

)()()(ˆ

)()(ˆ)(ˆ)(ˆ)()( 2

&&&&

&&&&

−⟩−⟨+

⟩−⟨+⟩−−−⟨+⟩−⟨+−∫−>−

Page 79: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Adaptive/Tracking: Simple RLC Model

∫ =+ 0])([ dtqqCqqL &&&&

iiRviqqRvqqCqLH

dyyCqLHq

)()]([)]([

)(2

1

0

2

−=−=+=

+= ∫&&&&&&

&

tvvqC

qRqL Ω==++ cos)(1

)( 0&&&

Goal:

Design applied voltage

that creates desired

limit cycle (e.g., 60 Hz)

∫ =− 0])([ dtiiRvi

∫ =+ 0])([ dtqqCqqL &&&&

Where storage terms matched

vi maximally used by load

EOM:

Hamiltonian and time derivative:

0)( >qRq && for 0≠q&

0)( >qCq for 0≠q

Page 80: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Control Design (cont.)

• Goal: 60Hz Limit Cycle

for matched storage terms and maximal real power

c) Specific Nonlinearities:

d) Tracking Hamiltonian with controller and information potentials:

[ ] [ ] 0)()( =−=+ ∫∫ dtqqRvdtqqCqLcc

&&&&&ττ

)(11 3 qsignRqRvq

Cq

CqL NL

NL

&&&& −−=

++

( ) ( )422 1

4

11

2

1)(

2

1R

NL

RCAPR qqC

qqKC

qqLH −+−

++−= &&

ΦΓΦ+ − ~~

2

1 1T

Icap VVVT +++=

Page 81: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Adaptive PID Control Strategy: Improve Performance

)(

~~]

1[

~~

2

1

2

11

2

1

2

1

1

1222

R

capRR

T

cap

Icap

qqe

vqReKqC

qLH

eKeC

eLH

VVVTH

−=

ΦΓΦ++−+−−=

ΦΓΦ+++=

+++=

&&&&&

&

RRR

R

qC

qRqLv

vvv

1ˆˆ ++=

∆+=

&&&

Controller

selected as:

Page 82: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

One Wants to Design for Ideal 60 Hz Condition

• Ideal

• Then

• Requires feedback control if

• To maintain a power factor of 1

• Select feedback portion as

2qRqv RR&& =

22 1Ω≠=

LCω

22 1Ω==

LCω

eKedKeKvt

dissgencap&∫ −−−=∆

Page 83: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Incorporate into Hamiltonian Time Derivative

∫ ΦΓ+Φ+−−= −t

TT

dissgen eYeeKedKH0

1 ]~

[~

][&

&&&& τ

][

]ˆ1ˆ[ˆ~

)]ˆ()11

()ˆ([~

)ˆ()11

()ˆ(~

qqqY

RC

L

RRCC

LL

qRRqCC

qLLY

RR

TT

T

RR

&&&

&&

&&&

&&&

=

=Φ=Φ

−−−=Φ

−+−+−=Φ

Where

Page 84: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Identify Adaptive Parameter Update Equations

)(

ˆ

1

ˆ

3

2

1

R

R

R

qqe

eqR

eqC

eqL

&&&

&&&

&

&

&&&&

−=

−=

−=

−=

γ

γ

γ

Page 85: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Perfect Parameter Matching: Nonlinear Stability Boundary

dteKdteedK diss

t

gen ∫∫∫ <′−ττ

τ 2

0][ &&

Results in asymptotically stable

(passivity) solution

Page 86: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Robustness/Tracking vs Adaptive/Tracking

5e65e6K_gen

10500K_diss

75010000K_cap

Gain Robustness Adaptive

Higher

controller

gains

achieved

Robustness

at expense of

increased

power usage

Page 87: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Linear RLC Network w/ PID Numerical Results

Page 88: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Linear RLC Network w/ PID/Adaptive Numerical Results

Page 89: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Linear RLC Network Energy & Power Flow Results

PID

Contr

oller

PID

/Adaptive

PID /adaptive

with

information

flow more

efficient than

PID with

physical flow

alone

Page 90: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Next Model Includes Nonlinearities

vqsignRqC

qC

qRqL NL

NL

=++++ )(11 3 &&&&

ADDED: Nonlinear Capacitance and Nonlinear Resistance

)()(

)(ˆ

][1

)(ˆˆ)(11ˆ

5

33

4

33

RR

NL

NL

NL

NL

RRref

qqeqqe

eqsignR

eeqC

qsignRqReqC

qC

qLv

&&&

&&&

&&

&

&&&&

−=−=

−=

−−=

++−++=

γ

γ

Page 91: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Nonlinear RLC Network w/ PID Numerical Results

Page 92: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Nonlinear RLC Network w/ PID/Adaptive Numerical Results

Page 93: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Nonlinear RLC Network Energy & Power Flow Results

PID

Contr

oller

PID

/Adaptive

PID /adaptive

with

information

flow more

efficient than

PID with

physical flow

alone

Page 94: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

PID/Adaptive Parameter Update Responses

Lin

ear

RLC

Model

Nonlinea

r R

LC

Model

Page 95: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Advanced Design Control (cont.)

Adaptive/Tracking Conclusions

•Applied new nonlinear power flow control design to power engineering

•Used to design adaptive PID control architecture

•The power flow and energy responses demonstrated the required energy consumption of each controller

•Adaptive PID control showed to be more efficient than just PID control alone

•Future work to extend apply to power grid control problems

Page 96: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

Page 97: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

1) Sinusoid Damping/ Nonlinear Feedback

i.) Model

Limit Cycles:

Where

for

)sin(xukxxm &&& ==+

0)sin( =+− xxx &&&

⇒>+= 02

1

2

1 22 xxH & Statically Stable

[ ] ( )[ ]xxxxxmH &&&&&& sin=+=

0)sin( == ∫ dtxxHcyclic&&&

τ

πnx ±=&,...3,1=n

Page 98: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

ii.) Nonlinear second-order model – responses for initial limit cycle

3D Hamiltonian 2D Phase Plane

Page 99: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Dynamically unstable, neutral stable, stable

Page 100: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

3D Hamiltonian 2D Phase Plane

• Nonlinear second-order model – responses for second limit cycle

Page 101: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Dynamically unstable, neutral stable, stable

(second limit cycle)

Page 102: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

3D Hamiltonian 2D Phase Plane

• Nonlinear second-order model – responses for BOTH limit cycles

Page 103: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

2) Multiple Input Multiple Output (MIMO)

i.) model:

12222

12111

FFxm

FFxm

+=

−=

&&

&&2

11 12

1xmH &=

2

2222

1xmH &=

Page 104: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

=1H& 111 xxm &&& [ ] 1121 xFF &−=

=2H& 222 xxm &&& [ ] 2122 xFF &+=

=12H 2

21

2

112

1

2

1xmxm && +

=12H& [ ] [ ] =+ 222111 xxmxxm &&&&&& )( 12122211 xxFxFxF &&&& −++

=INH2

1 2

1ii

N

i

xm &∑=

=INH& ii

N

i

xF &∑=1

[ ]jjjj

N

j

xxF && −+ ++

=

∑ 1)1(

1

1

Page 105: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

ii.) Decoupled (Neutral Stability) – Linear System Controller Design

Decoupling controller defined as:

=

=

=

−−−−=

−=

−=

=

+−−=

+

−==

2

1

2

1

2

1

0

0,,

ˆ

ˆ

,,0

0

12

12

12

12

12

12

0

2212

1211

2212

1211

2

1

122

121

I

I

I

D

D

D

P

P

P

t

DIP

K

KK

Kc

cKK

Kk

kKK

F

FxKxdKxKu

cc

ccC

kk

kkK

m

mM

uxCKxFF

FFFxM

&

&&&

τ

Page 106: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Decoupled (Neutral Stability)

2,12 =+

=+

= iforKc

K

m

Kk

i

ii

Dii

I

i

Pii

Page 107: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Decoupled

(Asymptotically Stable)

Page 108: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Decoupled (Unstable)

Page 109: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Decoupled Energy Responses (neutral, stable, unstable)

• Decoupled Power Flow Responses (neutral, stable, unstable)

Zero power flow for

dynamically neutral

stability decoupled

control case

Page 110: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

iii.) Coupled – Linear System Controller Design from combined Hamiltonian

Which matches up eigenvalues of system when (coupled integral gain matrix)

Eigenvalues and eignevectors:

[ ]

[ ][ ] [ ]

[ ] [ ]

[ ][ ] [ ][ ] 0

2

1

2

1

22

1

012

12

=Φ+−=Φ−+

++=

−+−=++=

++=

DIP

PDI

t

ID

T

P

T

P

TT

KCKMKK

KKMKCK

xdKxKCxxKKxMxH

xKKxxMxH

ωω

τ&&&&&&

&&

Page 111: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• System effectively decoupled into eigenspace:

Case Study #1:

Control Design Issues

Decoupled in

phase plane:

Page 112: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Similar results (Dissipative):

Page 113: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Similar results (Generative):

Page 114: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

iv.) Coupled – Linear System Controller Design where

With a single PID controller

The coupled Hamiltonian

Resulting time derivative

[ ]

( ) ( )

( ) ( )2121210

11112

2

1212

2

11

2

22

2

1112

01111

2121

11

1

111

2

1

2

1

2

1

2

1

00

xxcxdxKxKcH

xxkxKkxmxmH

xKdxKxKu

Fanduumm

t

ID

P

t

DIP

T

&&&&&

&&

&

−−

−+−=

−++++=

−−−=

==>

τ

τ

Page 115: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Coupled (Neutral)

Page 116: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Coupled (Asymptotically Stable)

Page 117: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Coupled (Unstable)

Page 118: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

v.) van der Pol / Nonlinear Controller:

1. Nonlinear MIMO system

2. Van der Pol damping treated as controller

3. Combined Hamiltonian

4. Time derivative

5. Neutral stability condition

uxxfKxxM ∆+−=+ ),( &&&

[ ] [ ]

[ ]∫∫ −==

−=+=

+=

ττdtxxfxdtH

xxfxKxxMxH

KxxxMxH

T

TT

TT

),(0

),(

2

1

2

1

12

12

12

&&&

&&&&&&

&&

( )( )

−=∆

−=∆

∆∆−+−

∆∆−−−=

12

12

12

2

2

2

2

2

1

2

1

ˆ

ˆ

)(

)(),(

121222

121211

F

Fu

xxx

xxKKxxKK

xxKKxxKKxxf

DGDG

DGDG

&&

&&&

Page 119: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Two-body dynamically neutral stability coupled control van der Pol system

•Positions

•Energy and

power flow

Page 120: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

• Two-body dynamically neutral stability coupled control van der Pol system

•Phase plane

projections

•Power flow

partitioned into

dissipation and

generation for

coupled 12

system

Page 121: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

3) Noncollocated Control

i.) Model

Case Study #1:

Control Design Issues

Page 122: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

0)()(

)()(

)()()()(

2222

21212121222

121122112111111

==

+−−−−=

+−−−−−−=

xfxg

uxxfxxgxm

uxxfxxgxfxgxm

&

&&&&

&&&&&

=1H )(2

111

2

11 xVxm +&

=1H& [ ] [ ] 11211221121111111 )()()()( xuxxgxxfxfxxgxm &&&&&&& +−−−−−=+

=2H2

222

1xm &

=2H& [ ] [ ] 2212121212222 )()( xuxxgxxfxxm &&&&& +−−−−=

Page 123: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

=12H ( ) ( )211211

2

22

2

112

1

2

1xxVxVxmxm −+++ &&

=12H& [ ] [ ] ))(()( 21211222211111 xxxxgxxmxxgxm &&&&&&&& −−+++

[ ] [ ] 2121222121121111 )()()( xxxgxmxxxgxgxm &&&&&& −++−++=

[ ] [ ] 22121211211211 )()()( xuxxfxuxxfxf &&&&&&& +−−++−−−=

[ ] [ ] [ ] )()()( 212112221111 xxxxfxuxuxf &&&&&&& −−−+++−=

Page 124: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

ii.) Design Controller

A) Pick and

Drive to !!

Statically Stable

0)( 11 >xV 0)( 2112 >− xxV

)0,0(

Page 125: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

B) Pick

−∫−−=

−∫−−=

222222

111111

2

1

xKdtxKxKu

xKdtxKxKu

DIp

DIp

&

&

=12H )()(2

1

2

1

2

1

2

1211211

2

2

2

1

2

22

2

11 11xxVxVxKxKxmxm pp −+++++ &

=12H& [ ] [ ] ))(()( 2111122222111111 21xxxxgxxKxmxxKxgxm pp&&&&&&&& −−+++++

[ ] [ ] 221212221121121111 21)()()( xxKxxgxmxxKxxgxgxm pp

&&&&& +−+++−++=

[ ] [ ] 22211111 221)( xdtxKxKxdtxKxKxf IDID

&&&&& ∫−−+∫−−−=

[ ] )()( 212112 xxxxf &&&& −−−+

Collocated

Control

Page 126: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

C) Pick

;2221 111xKdtxKxKu DIp&−∫−−= 02 =u

=12H )()(2

1

2

1211211

2

22

2

11 xxVxVxmxm −+++ &&

=12H& [ ] [ ] 2121222121121111 )()()( xxxgxmxxxgxgxm &&&&&& −++−++

[ ] [ ][ ] [ ][ ] [ ] )()()(

)()()(

)()()(

212112122211

2121121111

2121211211211

111xxxxfxxKdtxKxKxf

xxxxfxuxf

xxxfxuxxfxf

DIp&&&&&&&

&&&&&&

&&&&&&&

−−−+−∫−−−=

−−−++−=

−−++−−−=

Noncollocated

Control

Page 127: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

* ⇒−=∆ 12 xxx xxx ∆+= 12

=12H [ ] 111111 )()()()(11

xxxKdtxxKxxKxf DIp&&&& ∆+−∆+∫−∆+−−

[ ] xxf && ∆∆−+ )(12

[ ] [ ] xxfxxKdtxKxKxf DIp&&&& ∆∆−+−∫−−−= )()( 12111111 111

[ ] 1111xxKxdtKxK DIp&&∆−∆∫−∆−+

=⇒ 12H 2

1211211

2

22

2

11 12

1)()(

2

1

2

1xKxxVxVxmxm p+−+++ &

=12H& 11111 11111)( xxKxdtKxKxKxKxf

xu

DIpID&

44444 344444 21&&&

∆+∆∫+∆−∫−−−

∆∆

[ ] xxf && ∆∆−+ )(12How big is this?

Page 128: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

iii.) Collocated vs. NonCollocated (Unstable)

0 10 20 30 40-2

0

2

4

Position (m)

Time (sec)

x1

x2

x12

0 10 20 30 40-100

-50

0

50

Power Flow (W) Energy (J)

Time (sec)

H12 and Hdot 12

Diss

Gen

Dissdot

Gendot

0 10 20 30 40-10

-5

0

5

10

Position (m)

Time (sec)

x1

x2

x12

0 10 20 30 40-400

-200

0

200

400Power Flow (W) Energy (J)

Time (sec)

H12 and Hdot 12

Diss

Gen

Dissdot

Gendot

Page 129: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

iii.) Collocated vs. NonCollocated (Unstable)

0 10 20 30 400

20

40

60

Time (sec)

Energy (J)

H12

0 10 20 30 40-100

-50

0

50

Time (sec)

Power Flow (W)

Hdot12

0 10 20 30 400

10

20

30

40

50

Time (sec)

Energy (J)

H12

0 10 20 30 40-80

-60

-40

-20

0

20

Time (sec)

Power Flow (W)

Hdot12

Page 130: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

iii.) Collocated vs. NonCollocated (Unstable)

0 10 20 30 40-4

-2

0

2

4

Time (sec)

PID Control Effort (N)

u1

u2

0 10 20 30 40-100

-50

0

50

Time (sec)

PID Control Effort for ∆ x (N)

∆ u∆ x

0 10 20 30 40-10

-5

0

5

10

Time (sec)

PID Control Effort (N)

u1

u2

0 10 20 30 40-80

-60

-40

-20

0

20

Time (sec)

PID Control Effort for ∆ x (N)

∆ u∆ x

Page 131: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

iii.) NonCollocated (Neutral) vs. NonCollocated (Stable)

0 10 20 30 40-4

-2

0

2

4

Position (m)

Time (sec)

x1

x2

x12

0 10 20 30 40-150

-100

-50

0

50

100

150

Power Flow (W) Energy (J)

Time (sec)

H12 and Hdot 12

Diss

Gen

Dissdot

Gendot

0 10 20 30 40-4

-2

0

2

4

Position (m)

Time (sec)

x1

x2

x12

0 10 20 30 40-50

-40

-30

-20

-10

0

10

Power Flow (W) Energy (J)

Time (sec)

H12 and Hdot 12

Diss

Gen

Dissdot

Gendot

Page 132: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

iii.) NonCollocated (Neutral) vs. NonCollocated (Stable)

0 10 20 30 400

10

20

30

40

50

Time (sec)

Energy (J)

H12

0 10 20 30 40-80

-60

-40

-20

0

20

Time (sec)

Power Flow (W)

Hdot12

0 10 20 30 400

10

20

30

40

50

Time (sec)Energy (J)

H12

0 10 20 30 40-80

-60

-40

-20

0

20

Time (sec)

Power Flow (W)

Hdot12

Page 133: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

iii.) NonCollocated (Neutral) vs. NonCollocated (Stable)

0 10 20 30 40-4

-2

0

2

4

Time (sec)

PID Control Effort (N)

u1

u2

0 10 20 30 40-80

-60

-40

-20

0

20

Time (sec)

PID Control Effort for ∆ x (N)

∆ u∆ x

0 10 20 30 40-4

-2

0

2

4

Time (sec)

PID Control Effort (N)

u1

u2

0 10 20 30 40-80

-60

-40

-20

0

20

Time (sec)

PID Control Effort for ∆ x (N)

∆ u∆ x

Page 134: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

4) Stable / Unstable Systems

i.) Model : Nonlinear Linear Oscillator

)()( xfuxgxm &&& −=+

)(2

1 2 xVxmH += &

[ ] ( )[ ]xxfuxxgxmH &&&&&& −=+= )(

Equation of motion:

Page 135: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

ii.) Pick Functions:

=)(xV42

4

1

2

1xkkx NL+

=)(xf & )(xsigncxc NL&& +

=u xKxdtKxK DIp&−∫−−

( ) ( ) xdtKxsigncxKcxkxKkxm INLDNLp ∫−−+−=+++ )(3 &&&&

Page 136: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

iii.) Evaluate Control Design

A. Static Stability:

B. Dynamic Stability:

422

4

1)(

2

1

2

1xkxKkxmH NLp +++= &

⇒ Stable for 0, >NLkm and 0>+ pKk

⇒ Pick :0<+ pKk Unstable at )0,0(

( )[ ]xxdtKxsigncxKcH INLD&&&& ∫−−+−= )(

( )[ ] [ ] dtxxdtKxxsigncxKc INLDcc

&&&& ∫−=++ ∫∫ ττ)(

Limit cycle:Gain Scheduling

Page 137: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

0>+ pKk

:0<+⇒ pKk Unstable at )0,0(

Stable for⇒

Page 138: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1:

Control Design Issues

5) Many popular coupled dynamical systems: (robots,

spacecraft, etc.) can be modeled as

External torque

Generalized position

Generalized velocity

Symmetric mass matrix

Centripetal/Coriolis vector

Gravitational vector=

=

=

=

=

=

++=

)(

),(

)(

)(),()(

qG

qqC

qM

q

q

qGqqCqqM

&

&

&&&

τ

τ

Dynamical

systems in

this form can

be shown to

be designed as

decoupled

SISO systems

Work rate of the gyroscopic or

centripetal/Coriolis terms are

zero [Ref. Slotine, Robinett]

[Ref. Slotine] J..-J.E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, Inc., N.J., 1991

[Ref. Robinett] R.D. Robinett, G.G. Parker, H. Schaub, J.L. Junkins, Lyapunov Optimal Saturated Control for

Nonlinear Systems, J. Guidance, Control, and Dynamics, Vol. 20, No. 6, pp. 1083-1088, Nov.-Dec., 1997

Page 139: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1: MIMO 2 DOF

Planar Robot w/ PID Control System

qqq

qKdqKqKqCqH

qqCqqH

ref

t

DIPrefref

−=

−−−+=

=+

∫~

~~~ˆˆ

),()(

0

&&&&

&&&

ττ

τEOM and PID Control Law:

Page 140: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1: Lyapunov Function/

Hamiltonian Based on Error Energy

[ ] [ ]

ave

t

I

T

aveD

T

aveioave

t

I

T

D

T

P

TT

dqKqqKq

STW

dqKqqKq

qKqqHqH

]~~[]~~[

~~~~

~~

2

1~~

2

1

0

0

−=

∆=∆

−−=

+=∆=

τ

τ

&&&

&&

&&&&

&&

VVVV

VVVV

Nonlinear stability boundary

Applied per DOF

(assumes perfect

parameter matching)

Page 141: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #1: Planar Robot w/ PID Control

Numerical Simulation Results - Case 1

0 1 2 3 4 5-2

-1

0

1

2

3

Time (sec)

Position (rad)

q1ref

q1

q2ref

q2 0 1 2 3 4 5

-25

-20

-15

-10

-5

0

5

Time (sec)

Diss\Gen Exergy (J) and Exergy Rate (W)

To ∆ SDOT

∆ WDOT

To ∆ S

∆ W

0 1 2 3 4 5-40

-30

-20

-10

0

10

Time (sec)

Diss\Gen Exergy (J) and Exergy Rate (W)

To ∆ SDOT

∆ WDOT

To ∆ S

∆ W

DOF 1

DOF 2

DISSIPATIVE

Page 142: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

0 1 2 3 4 5-3

-2

-1

0

1

2

3

Time (sec)

Position (rad)

q1ref

q1

q2ref

q2

0 1 2 3 4 5-300

-200

-100

0

100

200

300

Time (sec)

Diss\Gen Exergy (J) and Exergy Rate (W)

To ∆ SDOT

∆ WDOT

To ∆ S

∆ W

0 1 2 3 4 5-50

0

50

Time (sec)

Diss\Gen Exergy (J) and Exergy Rate (W)

To ∆ SDOT

∆ WDOT

To ∆ S

∆ W

DOF 1

DOF 2

NEUTRAL STABILITY

Case Study #1: Planar Robot w/ PID Control

Numerical Simulation Results - Case 2

Page 143: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

0 1 2 3 4 5-10

-5

0

5

10

15

Time (sec)

Position (rad)

q1ref

q1

q2ref

q2

0 1 2 3 4 5-6

-4

-2

0

2

4

6x 10

4

Time (sec)

Diss\Gen Exergy (J) and Exergy Rate (W)

To ∆ SDOT

∆ WDOT

To ∆ S

∆ W

0 1 2 3 4 5-6

-4

-2

0

2

4

6

8x 10

4

Time (sec)

Diss\Gen Exergy (J) and Exergy Rate (W)

To ∆ SDOT

∆ WDOT

To ∆ S

∆ W

DOF 1

DOF 2

GENERATIVE

Case Study #1: Planar Robot w/ PID Control

Numerical Simulation Results - Case 3

Page 144: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #2

Collective Plume Tracing:

A Minimal Information Approach

to Collective Control

Page 145: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• Problem: Track a chemical plume to its source; find buried landmines

• Characteristics: 1- Nonlinear time/spatially varying plume field

2-Complicated model-turbulence

3-Difficult to correlate model to data in time/space

4- Can’t count on wind or water flow direction

5- Sensor Latency

Solution ??

),( txC

Case Study #2:

Collective Plume Tracing

Page 146: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• Possible solution: Person with a chemical sensor and a GPS receiver

1. Time–spatial correlation issues

2. Sensor latency/dynamic range issues

3. No redundancy

4. Leads to exhaustive search

Case Study #2:

Collective Plume Tracing

Page 147: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• Collective Solution: Team of simple robots performing a collective search

– Distributed sensing

– Decentralized control (optimization/Newton update)

– Simple model of plume

– Redundant

– Independent of latency for stability (Performance?)

– Minimize processing, memory, and communications

– Beat down noise

x

x

x

x

x

x

r

Quadratic FitConcentration

Case Study #2:

Collective Plume Tracing

Page 148: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• Emergent Behavior: Quadratic fit w/o memory

• Evaluate Resource Utilization: Trade-off between processing,

memory, and communications

– Baseline: Minimize all three simultaneously

• 8 bit processor, no memory, broadcast 3 words

Minimum

Norm

Exact Fit

Least squares

# of robots

Surface

Fit

Index

6

Case Study #2:

Collective Plume Tracing

Page 149: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• Goal of DARPA Distributed Robotics [Ref. Byrne] Program: build smallest, dumbest robot that couldn’t do anything other than random motion

• As a team of like robots - could solve complicated problem

• N land-based robots whose task to localize a source that emits a measurable scalar field F(x)

• Robots evaluate the field with a sensor and communicate locations and sensor measurements to neighboring robots

• Robots shared sensor information, but individually decided course of action based on their own estimate of plume field

• Robot samples its environment, broadcasts information to others

0.75x0.71x1.6 inches

Onboard temperature sensor

8-bit RISC processor

CSMA radio network

50 Kbits/sec data rate

[Ref. Byrne] Byrne, et. al., Miniature Mobile Robots for Plume Tracking and

Source Localization Research, J. MicroMech., Vol. 1, No. 3, 2002.

Case Study #2:

Collective Plume Tracing

Page 150: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• Universal Mathematical Approach

– Nonlinear control design via exergy/entropy thermodynamic concepts

– Hamiltonian (universal function) is the key element

– Each column has the ability to work on collective systems

Hamiltonian

Lyapunov

FunctionalsStatistical

Mechanics

Nonlinear

Control

Theory

Equations

of State

Quantum

Mechanics

State

Functions

Chemical

Kinetics

Chemical

Potential

Calculus of

Variations

Equations

of Motion

Shannon

Information

Fisher

Information

Case Study #2:

Collective Plume Tracing

Page 151: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• Kinematic Control – No dynamics; assume tight autopilot loop; control velocity (speed)

1- Need to measure plume field: concentration at a point in time

2- Convert concentration measurements to range and bearing to target

Fit a quadratic surface to distributed sensor measurements

Perform decentralized Newton updates for feedback control

3- Shape the “virtual potential” with a minimum (maximum) at

for Hamiltonian to meet static stability requirements for the robot:

−),( txC

⇒⇒

),( txcxcxxcctxc T

o 21ˆ

2

1ˆˆ),(ˆ +++=

kk xccxx

c+−=⇒=

∂ −+ 1

1

21ˆˆ0

ˆα

x1

1

2ˆˆ* ccx −−=

thi

0ˆ2

1ˆˆˆˆ

2

1)( 211

1

21 >++== −i

T

ii

TT

ici xcxxccccxVHi

Case Study #2:

Collective Plume Tracing

Page 152: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

And collective robots

4-Design the power flow, time derivative of the Hamiltonian, to meet the

dynamic stability requirements for the feedback controller

011

>== ∑∑==

ic

N

i

i

N

i

VHH

[ ] iiii xxxccxcccx −=−−=+−= −− *ˆˆˆˆˆ1

1

221

1

2& (Linear tracker)

[ ] [ ] [ ]

0

0ˆˆˆˆˆˆˆ

1

21

1

22121

<=

<++−=+=

∑=

i

N

i

i

T

ii

T

ii

HH

xcccxccxccxH

&&

&&

[ ] **

11

11xxxx

Nxx

Nx cmi

N

i

cmi

N

i

cm +−=+−=⇒= ∑∑==

&

Case Study #2:

Collective Plume Tracing

Page 153: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• Kinetic Control – dynamics included:

1-Shape the Hamiltonian surface to meet the static stability requirements for the

robot

and the collective of robots

2- Design the power flow, time derivative of the Hamiltonian, to meet the

dynamic stability requirements for the feedback controller

iii uxm =&&

thi 02

1>+=+=

ii cii

T

icii VxMxVTH &&

[ ] 011

>+== ∑∑==

ici

N

i

i

N

i

VTHH

x

Vu ic

i∂

∂−=

iDiI xKdtxK &−∫−

[ ]

0

0

1

<=

<−∫−=

∂+=

∑=

i

N

i

i

iDiI

T

i

c

ii

T

ii

HH

xKdtxKxx

VxMxH i

&&

&&&&&&

Case Study #2:

Collective Plume Tracing

Page 154: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #2: Collective Robots

Numerical Results

Convergence:

Dissipative > Generative

for the collective

Page 155: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #2: Collective Robots

Stability Boundary

• Collective robot oscillations at the limit cycle

Page 156: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• Information Theory Applied To:

1- Kinematic Control: Shannon information /entropy

a) Fundamental trade-off: processing, memory, and communications

minimize all three simultaneously

8 bit processor, no memory, 3 words to communicate

b) Stability: latency independent due to no dynamics; stop until next

update/command

c) Performance: the limitation is the sensor update rate; 60 sec

Channel/system capacity of an equiprobable source

=aveC

( )

( )sec1.080log

60

1

;805.0/40/

2

bitC

etemperaturTbitsTTTn

ave

LOWHIGH

<=⇒

−==∆−=COMM. Link = 50

Kbits/sec

Case Study #2:

Collective Plume Tracing

aveHbits

n &==sec

log1

informationtime

Page 157: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• Kinetic Control: Fisher Information

a) Fisher Information – Measure of how well the receiver can estimate the

message from the sender.

Shannon Entropy – Measure of the sender’s transmission efficiency

over a communications channel

b) Hamiltonian is exergy; portion of energy that can do work; physical and

information exergies

c) Fisher Information:

“Real Amplitude” function of the probability

density function

dxx

xqdx

x

xp

xpdxxp

x

xpI

222)(

4)(

)(

1)(

)(ln

∂∫=

∂∫=

∂∫=

−= )()(2 xpxq

Case Study #2:

Collective Plume Tracing

Page 158: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

d) ‘Mean Kinetic Energy” interpretation of Fisher Information

From Quantum Mechanics in the “Classical Limit” of the expectation of

the momentum squared

Tdtm

dtqI2

44 2 ∫=∫= &

dxx

tx

mp

222 ),(

2 ∂

Ψ∂∫=

h

Case Study #2:

Collective Plume Tracing

Page 159: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

- Dirac’s constant

- wave function

- expectation operator

The classical limit is reached when

Is equivalent to

),( txΨ

h

tt

ttxF

dx

xdVx

dt

dmp

dt

d)(

)(2

2

=−==

Case Study #2:

Collective Plume Tracing

)()(

2

2

classical

classical

classicalclassical xF

dx

xdVx

dt

dm =−=

Page 160: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Which occurs when

and

is small

This occurs when

which leads to

( ) ( ) ( )xFxxxFxF ′−+≈

( )xF≈

Case Study #2:

Collective Plume Tracing

xxclassical =

( ) ( )22xxx −=∆

( ) ( ) ( ) ( ) ( ) ...!2

1)(

2+′′−+′−+= xFxxxFxxxFxF

Page 161: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

e) Fisher Information Equivalency and Exergy

where

i

i

N

i

Ic Hm

VVVTH1~~~~~

1

∑=

=+++=

i

i

N

i

Tm

T1~

1

∑=

=

==∑=

j

j

N

j

Vm

V1~

1

==∑=

kc

k

N

k

c Vm

V1~

1

potential

control potential

==∑=

lI

l

N

l

I Vm

V1~

1

information potential

Case Study #2:

Collective Plume Tracing

Page 162: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

( )[ ]dtVVVTdtHJI Ic

~~~~8

~8 +++∫=∫=+

0~

>=+ HJI &&

0~

<=+ HJI&&&&&

( ) =++∫= dtVVVJ Ic

~~~8Where

(Static stability)

(Dynamic stability)

Bound Fisher Information

Case Study #2:

Collective Plume Tracing

[Ref. Frieden] B.R. Frieden, Physics from Fisher Information: A Unification,

Cambridge University Press, 1999.

Page 163: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #2:

Summary and Conclusions

•Analyze and design “emergent” behaviors to enable a team

of simple robots to perform plume tracing with assistance of

information theory

•Demonstrated fundamental nature of Hamiltonian function

in design of collective systems

• Equivalences between physical and information-based

exergies shown for Shannon information, Fisher

information, and virtual fields

• Fisher Information Equivalency developed that can serve as

an ideal optimization functional to measure performance and

stability of collective system with respect to required

information resources

Page 164: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3

Nonlinear Aeroelasticity

Page 165: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Aero-Servo-Elasticity

Coupling

Aerodynamics

Turbulence

Wind gusts

Flutter

Dynamic Stall

Structures

Multi-Dynamics

Elasticity

Flutter

Lightweight

Controls

Classical/Modern

Nonlinear

Safety

Reliability

Intelligent

Case Study #3:

Challenging Engineering Problem

Page 166: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3:

Challenging Engineering Problem

Exergy/Entropy Nonlinear Power Flow Control:

• Rigid Torque Controller Region II & II ½

• Hybrid Pitch and Active Aero Controller Region III

• Include: Dynamic Stall, Classical Flutter, etc.

WINDPACT 1.5 MW WT:

Region III starts around 12 m/s

Page 167: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

WT Plant

Wind Conditions

Sensor

Feedback• Tip Deflection

• Tip Rate

• Tip Accelerometer

• Other

Active Aero

Actuator Device

• MicroTabs

• Morphing Wing

• Conventional Flaps

Control

System

Case Study #3:

Active Aero Load Control Architecture Analysis/Design

Model

Reference

Control

System

Optimized Energy Capture

• Region II

• Region II ½

• Region III

• Conventional

(PD,PID,PIDA)

• Rate Feedback

• PPF

• SMC

• State-Space

• Nonlinear Power Flow

Page 168: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3:

Nonlinear Aeroelasticity

• Wind tunnel model experienced stall flutter over its entire flight envelope M=0.4-0.9;

-10oto -3

oand 10

oto 3

oangle of attack [Ref. Guiterrez]

•The wings broke and cracked at the 1/3 span position due to

fatigue; first torsional mode ~ 450 Hz

[Ref. Guiterrez] W.T. Gutierrez, R. Tate, H. Fell, Investigation of a

Wind Tunnel Model High Aspect Ratio Wing Fracture, AIAA 94-

2558, 18th AIAA Aeropsace Ground Testing conference, June 1994,

Colorado Springs, CO.

Page 169: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3:

Nonlinear Aeroelasticity

• Nonlinear aerodynamic and structural model

– Nonlinear stall flutter model

– Single DOF model

Page 170: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3:

Nonlinear Aeroelasticity

- Equations of motion derived from Lagrange’s equation:

αταα

ααα

αα

αα

α

αα

αα

α

α

&

&

&&

&

&

&

D

t

Ipcontrol

aero

NLdamp

controlaerodamp

NL

KdKKuQ

MMQ

signCCQ

QQQQ

KKV

IT

VTL

where

QLL

dt

d

−−−==

+=

−−=

++=

+=

=

−=

=∂

∂−

∫0

42

2

),()(

)(

4

1

2

1

2

1

Page 171: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3:

Nonlinear Aeroelasticity

=0

ˆ),(

ααα α

α

&& &

&MCM

=

0

0

ˆ

)(

α

αα

α

MC

M

stall

stall

αα

αα

>

<

- The aerodynamic moments are generated based on the following hysteresis logic:

for

for

for the return hysteresis cycle

for

for

stall

stall

αα

αα

>

<

- Applying Lagrange’s equation yields the EOM :

),()()(3 αααααααα αα&&&&&

&MMusignCCKKI NLNL +++−−=++

Page 172: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

A) Linear Region

For the model is linear with or

which produces typical linear aeroelastic behavior. Divergence occurs when

where

Torsional flutter occurs when

where

,stallαα < 0== NLNL CK

KCM ≥α

ˆ for 0=u

[ ] [ ] uCKCCI MM =−+−+ ααααα

ˆˆ &&&&

== 2

2

1ˆ VAKqAKC MMM ρααα

CCM ≥α&

ˆ for 0ˆ >−αMCK

and 0=u

.ˆ qAdKC MM αα &&=

Case Study #3:

Nonlinear Aeroelasticity

Page 173: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

B) Nonlinear Stall Flutter

When the motion reaches the model becomes nonlinear where

with

which produces

,stallαα >

),()( αααααα αα&&&&

&MMKCI +=++

22

2

1

2

1αα KIH += &

[ ] [ ]αααααααα αα&&&&&&&

& ),()( MMCKIH ++−=+=

[ ] [ ] dtCdtMM ααααααταατ

&&&&& ∫∫ =+ ),()(

Case Study #3:

Nonlinear Aeroelasticity

Page 174: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3:

Nonlinear Stall Flutter Linear Dynamics Results

-40

-20

0

20

40

-40

-20

0

20

40

0

0.5

1

1.5

2

2.5

α (deg)

H = 0.5*I*α dot2 + 0.5*k*α

2 + 0.25*k

NL α4

α dot (deg/s)

Hamiltonian (J)

Dissipate

Neutral

Generate

-40 -20 0 20 40-40

-20

0

20

40

α dot (deg/sec)

α (deg)

Phase Plane

Case 1

Case 2

Case 3

Page 175: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3:

Nonlinear Stall Flutter Linear Dynamics Results

0 5 10 15 20-2

0

2

4

6

angle (deg) angle rate (deg/sec)

Time (sec)

CASE 1 Passive

α

α dot

0 5 10 15 20-40

-20

0

20

40

angle (deg) angle rate (deg/sec)

Time (sec)

CASE 3 Generative

α

α dot

0 5 10 15 20-15

-10

-5

0

5

10

15

angle (deg) angle rate (deg/sec)

Time (sec)

CASE 2 Neutral

α

α dot Resp

onses

0 5 10 15 20-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

Time (sec)

Power Flow (W) and Energy (J)

Case 1 Dissipative

Pgen

Egen

Pdiss

Ediss

0 5 10 15 20-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec)

Power Flow (W) and Energy (J)

Case 2 Neutral

Pgen

Egen

Pdiss

Ediss

Pow

er Flo

w

0 5 10 15 20-6

-4

-2

0

2

4

6

8

Time (sec)

Power Flow (W) and Energy (J)

Case 3 Generative

Pgen

Egen

Pdiss

Ediss

Page 176: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3:

Nonlinear Stall Flutter Linear Dynamics Results

Aero Moments

-2 0 2 4 6-0.2

0

0.2

0.4

0.6

0.8

α (deg) and α dot (deg/s)

Aero Moments (N-m)

Case 1 Dissipative: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

-20 -10 0 10 20-2

-1

0

1

2

α (deg) and α dot (deg/s)

Aero Moments (N-m)

Case 2 Neutral: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

-40 -20 0 20 40-10

-5

0

5

10

α (deg) and α dot (deg/s)

Aero Moments (N-m)

Case 3 Generative: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

Page 177: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

C) The nonlinear stall flutter can be further modified by adding the nonlinear

stiffness and damping

with

which produces a limit cycle when

),()()( 3 αααααααα αα&&&&&

&MMKKsignCCI NLNL +=++++

[ ] [ ] dtsignCCdtMM NL αααααααταατ

&&&&&& )(),()( +=+ ∫∫

422

4

1

2

1

2

1ααα NLKKIH ++= &

[ ] [ ]αααααααααα αα&&&&&&&&

& ),()()(3 MMsignCCKKIH NLNL ++−−=++=

Case Study #3:

Nonlinear Aeroelasticity

Page 178: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3:

Nonlinear Stall Flutter NonLinear Dynamics Results

-40

-20

0

20

40

-30

-20

-10

0

10

20

30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

α (deg)

H = 0.5*I*α dot2 + 0.5*k*α

2 + 0.25*k

NL α4

α dot (deg/s)

Hamiltonian (J)

Dissipate

Neutral

Generate

-40 -20 0 20 40-30

-20

-10

0

10

20

30

α dot (deg/sec)

α (deg)

Phase Plane

Case 1

Case 2

Case 3

Page 179: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3:

Nonlinear Stall Flutter NonLinear Dynamics Results

Resp

onses

Pow

er Flo

w

0 5 10 15 20-1

0

1

2

3

4

5

angle (deg) angle rate (deg/sec)

Time (sec)

CASE 1 Passive

α

α dot

0 5 10 15 20-15

-10

-5

0

5

10

15

angle (deg) angle rate (deg/sec)

Time (sec)

CASE 2 Neutral

α

α dot

0 5 10 15 20-30

-20

-10

0

10

20

30

angle (deg) angle rate (deg/sec)

Time (sec)

CASE 3 Generative

α

α dot

0 5 10 15 20-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

Time (sec)

Power Flow (W) and Energy (J)

Case 1 Dissipative

Pgen

Egen

Pdiss

Ediss

0 5 10 15 20-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec)

Power Flow (W) and Energy (J)

Case 2 Neutral

Pgen

Egen

Pdiss

Ediss

0 5 10 15 20-4

-2

0

2

4

6

Time (sec)

Power Flow (W) and Energy (J)

Case 3 Generative

Pgen

Egen

Pdiss

Ediss

Page 180: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #3:

Nonlinear Stall Flutter Linear Dynamics Results

Aero Moments

-2 0 2 4 6-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

α (deg) and α dot (deg/s)

Aero Moments (N-m)

Case 1 Dissipative: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

-20 -10 0 10 20-2

-1

0

1

2

α (deg) and α dot (deg/s)

Aero Moments (N-m)

Case 2 Neutral: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

-40 -20 0 20 40-6

-4

-2

0

2

4

6

α (deg) and α dot (deg/s)

Aero Moments (N-m)

Case 3 Generative: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

Page 181: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

D) The nonlinear system can be modified by feedback control to meet several

performance requirements. A PID controller is implemented to show the

effects of feedback control. The model becomes

with

which produces a limit cycle when

[ ] [ ] τααααααααα αα dKMMsignCKCKKKIt

INLDNLp ∫−++−+−=+++0

3 ),()()( &&&&&&

[ ] 422

4

1

2

1

2

1ααα +++= pKKIH &

[ ]αααα &&&& 3)( NLp KKKIH +++=

αταααααα αα&&&&

&

−++−+−= ∫ dKMMsignCKC

t

INLD0

),()()()(

( )[ ] dtsignCKCdtdKMM NLD

t

I αααατααααταατ

&&&&&& )(),()( 1

0++=

−+ ∫∫∫

Case Study #3:

Nonlinear Aeroelasticity

Page 182: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

-60-40

-200

2040

60

-100

-50

0

50

100

0

1

2

3

4

5

6

7

8

9

α (deg)

H = 0.5*I*α dot2 + 0.5*k*α

2 + 0.25*k

NL α4

α dot (deg/s)

Hamiltonian (J)

Dissipate

Neutral

Generate

-100 -50 0 50 100-100

-50

0

50

100

α dot (deg/sec)

α (deg)

Phase Plane

Case 1

Case 2

Case 3

• Nonlinear Power Flow Control Design Dynamic Stall: Limit Cycle Identification

Case Study #3:

Dynamic Stall Control System Results

Page 183: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

0 5 10 15 20-30

-20

-10

0

10

20

30

angle (deg) angle rate (deg/sec)

Time (sec)

CASE 1 Passive

α

α dot

0 5 10 15 20-40

-20

0

20

40

angle (deg) angle rate (deg/sec)

Time (sec)

CASE 2 Neutral

α

α dot

0 5 10 15 20-100

-50

0

50

100

angle (deg) angle rate (deg/sec)

Time (sec)

CASE 3 Generative

α

α dot

Resp

onses

Case Study #3:

Dynamic Stall Control System ResultsPow

er Flo

w

0 5 10 15 20-4

-3

-2

-1

0

1

2

3

Time (sec)

Power Flow (W) and Energy (J)

Case 1 Dissipative

Pgen

Egen

Pdiss

Ediss

0 5 10 15 20-15

-10

-5

0

5

10

15

Time (sec)

Power Flow (W) and Energy (J)

Case 2 Neutral

Pgen

Egen

Pdiss

Ediss

0 5 10 15 20-40

-20

0

20

40

Time (sec)

Power Flow (W) and Energy (J)

Case 3 Generative

Pgen

Egen

Pdiss

Ediss

Page 184: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Aero Moments

Case Study #3:

Dynamic Stall Control System Results

-40 -20 0 20 40-5

-4

-3

-2

-1

0

1

2

α (deg) and α dot (deg/s)

Aero Moments (N-m)

Case 1 Dissipative: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

-40 -20 0 20 40-10

-5

0

5

10

α (deg) and α dot (deg/s)

Aero Moments (N-m)

Case 2 Neutral: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

-100 -50 0 50 100-15

-10

-5

0

5

10

15

20

α (deg) and α dot (deg/s)

Aero Moments (N-m)

Case 3 Generative: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

Page 185: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

•Nonlinear power flow control was applied to a

nonlinear stall flutter problem (dynamic stall)

•Nonlinear structural and discontinuous aerodynamic

models were directly accommodated

•The limit cycles were found by partitioning power

flows

•The limit cycles were shown to be stability boundaries

•Flutter suppression controllers were initially assessed

Case Study #3:

Summary and Conclusions

Page 186: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #4

Power Engineering

Page 187: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #4:

Power Engineering - OMIB

I. Model (Simple one-machine infinite bus swing equation):

II. Controller Design:

( )[ ]

( ) δδδ

ωδ

ωδ

ωδω

ωδ

&&&

&

&&&

&

&

DPPM

DPPM

mMAX

MAXm

−=+

=

=

−−=

=

sin

sin1

( )[ ]

( )[ ] [ ]

[ ] 0

0

sin

cos12

1 2

<++−⇒

<⇒

++−=+=

−+=

∫ δδ

δδδδδ

δδ

τ

&&

&

&&&&&&

&

uPD

H

uPDPMH

PMH

m

mMAX

MAX

( )

0

0547.1

)sin(0

==

=

−−−−−= ∫

DP

s

t

sIDsP

KK

dKKKu

δ

τδδδδδ &

Page 188: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #4:

Power Engineering- OMIB Trajectories

-8 -6 -4 -2 0 2 4 6 8-10

-5

0

5

10

0

1

2

3

4

0.5 8/(100 3.1415) (ω)2+1.2647 (1-cos(δ))

δ

ω

• Dark blue (2.0865,0)

•Black (2.0865*1.2,0)

•Green (2.0865*1.2,0), K_I = 1.0

•Cyan (-4.197,0)

•Red (-4.196,0)

•Magenta (-4.197,0), K_I = 0.01

•Magenta (dash) (-4.197,0), K_I = 0.05

Page 189: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #4:

Power Engineering - 2MIB

I. Model (Two-machine infinite bus swing equations):

II. Controller Design:

[ ]

[ ]222232112

2

2

111131212

1

1

22

11

sinsin1

sinsin1

2

1

ωδδω

ωδδω

ωδ

ωδ

DCCPM

DCCPM

m

m

−−−=

−−−=

=

=

&

&

&

&

[ ] [ ] [ ]

[ ] [ ]( )

( ) 212122222322

121121111311

2222111112

2112223113

2

22

2

1112

sinsin

sinsin

cos1cos1cos12

1

2

1

2

1

21

uCDPCM

uCDPCM

uDPuDPH

CCCMMH

m

m

mm

+−−−=+

+−−−=+

+−++−=

−+−+−++=

δδδδδ

δδδδδ

δδδδ

δδδδδ

&&&

&&&

&&&&&

&&

For:

Page 190: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #4:

Overview of Lanai “Like” Model

• 12470 V, 5.5 MW peak load

• 4 Diesel Generators (2 – 2.2 MW, 2 – 1.0 MW) with controls

• 1.2 MW High Level PV controls

• Switchable Loads

• Distribution lines (series RL) calculated for 350kcmil under 5 miles

• 3-phase faults at each bus

Model Courtesy: Ben Schenkman

Lanai “Like” Model Parameter Characteristics:

• Real World Microgrid – “Lanai and Kauai”

• “Like” Model of Real World Microgrid

• Advanced Controls on Distributed Generation

Page 191: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #4:

Lanai ”Like” MicroGrid Model

Model Courtesy: Ben Schenkman

Page 192: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov
Page 193: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

YELLOW – Electrical Power Output

PURPLE – Control Output

BLUE – Speed Error

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Page 195: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov
Page 196: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #4:

Lanai “Like” Control Model

•Nonlinear Power Flow Control Design:

∑4

1

generators

PV

Generation

function (sun)

Loads setup similar to

Lanai “like” model

Page 197: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

Case Study #4:

Lanai “Like” Control Model

0 0.1 0.2-0.4

-0.2

0

0.2

0.4

Charge (mA-sec)

Time (sec)

qref

q

0 0.1 0.2-100

0

100

200

Charge rate (mAmp)

Time (sec)

qdot

ref

qdot

0 0.1 0.2-100

-50

0

50

100

Vin (Volts)Time (sec)

v

vref

∆ v

videal

0 0.05 0.1 0.15 0.2-5

0

5

10

Rhat (Ohm)

Time (sec)

0 0.1 0.2-0.4

-0.2

0

0.2

0.4

Energy (mJoules)

Time (sec)

∫ Pdiss d τ

∫ Pgen d τ

0 0.1 0.2-0.4

-0.2

0

0.2

0.4

Power Flow (Watts)

Time (sec)

Pdiss

Pgen

Page 198: Nonlinear Power Flow Control Designwindpower.sandia.gov/other/088055.pdf · Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov

• Investigated OMIB model – Limit cycles and control

design options

• Investigated 2MIB model

• Identified microgrid model based on actual Lanai

“like” electric power grid system

•Evaluated Exergy/entropy control design for

simplified Lanai “like” model

Case Study #4:

Summary and Conclusions


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